gas dynamics - astrophysics
TRANSCRIPT
Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit
This course:
Lectures on Monday, HFML0220; 13.30-15.30; Assignment course (werkcollege): Friday, HG00.065, 08:30-10:30;
Lecture Notes and PowerPoint slides on: www.astro.ru.nl/~achterb/Gasdynamica_2015
Overview What will we treat during this course? • Basic equations of gas dynamics - Equation of motion - Mass conservation - Equation of state
• Fundamental processes in a gas - Steady Flows - Self-gravitating gas - Wave phenomena - Shocks and Explosions - Instabilities: Jeans’ Instability
Applications
• Isothermal sphere & Globular Clusters
• Special flows and drag forces • Solar & Stellar Winds • Sound waves and surface waves on water • Shocks • Point Explosions, Blast waves & Supernova Remnants
LARGE SCALE STRUCTURE
Classical Mechanics vs. Fluid Mechanics
Single-particle (classical) Mechanics
Fluid Mechanics
Deals with single particles with a fixed mass
Deals with a continuum with a variable mass-density
Calculates a single particle trajectory
Calculates a collection of flow lines (flow field) in space
Uses a position vector and velocity vector
Uses a fields : Mass density, velocity field....
Deals only with externally applied forces (e.g. gravity, friction etc)
Deals with internal AND external forces
Is formally linear (so: there is a superposition principle for solutions)
Is intrinsically non-linear No superposition principle in general!
Basic Definitions
Mass, mass-density and velocity
Mass ∆m in volume ∆V Mean velocity V(x , t) is defined as:
Mass density ρ:
Equation of Motion: from Newton to Navier-Stokes/Euler
Particle α
Equation of Motion: from Newton to Navier-Stokes/Euler
Particle α You have to work with a velocity field that depends on position and time!
Derivatives, derivatives…
Eulerian change: fixed position
Derivatives, derivatives…
Eulerian change: evaluated at a fixed position
Lagrangian change: evaluated at a shifting position
Shift along streamline:
t∂∂
d Vdt t
∂= + •∇∂
x
y
z
Gradient operator is a ‘machine’ that converts a scalar into a vector:
Related operators: turn scalar into scalar, vector into vector….
( ) ( ) 2
scalar into vector:
vector into scalar: 4
4vector into vector:
tensor into vector:
Useful relations: = 0 , 0 ,
G
c
π ρ
π
= − Φ
• = −
× =
• = −
• × × Φ = • Φ = ∇ Φ
T
g
g
B J
f
B
1. Define the fluid acceleration and formulate the equation of motion by analogy with single particle dynamics;
2. Identify the forces, such as pressure force;
3. Find equations that describe the response of the other fluid properties (such as: density r, pressure P, temperature T) to the flow.
Equation of motion for a fluid:
Equation of motion for a fluid:
d ( )dt t
∂≡ = + •
∂V Va V V
The acceleration of a fluid element is defined as:
Equation of motion for a fluid:
This equation states: mass density × acceleration = force density note: GENERALLY THERE IS NO FIXED MASS IN FLUID MECHANICS!
Equation of motion for a fluid:
Non-linear term! Makes it much more difficult To find ‘simple’ solutions. Prize you pay for working with a velocity-field
Equation of motion for a fluid:
Non-linear term! Makes it much more difficult To find ‘simple’ solutions. Prize you pay for working with a velocity-field
Force-density This force densitycan be: • internal: - pressure force - viscosity (friction) - self-gravity • external - For instance: external gravitational force
Split velocities into the average velocity
V(x, t),
and an
isotropically distributed deviation from average,
the random velocity:
σ(x, t)
Effect of average over many particles in small volume:
For isotropic fluid:
Vector
Three notations for the same animal!
Vector
Scalar
Rank 2 tensor
Rank 2 tensor T
Vector
This is the product rule for differentiation!
Tensor divergence:
Isotropy of the random velocities:
Second term = scalar x vector! This must vanish upon averaging!!
Isotropy of the random velocities
Diagonal Pressure Tensor
Equation of motion for frictionless (‘ideal’) fluid:
• We know how to interpret the time-derivative d/dt; • We know what the equation of motion looks like;
• We know where the pressure force comes from (thermal motions), and how it looks: f = - ÑP . • We still need:
- A way to link the pressure to density and temperature: P = P(r, T); - A way to calculate how the density r of the fluid changes.